An Approximation to the Boussinesq Equations

Transcription

1 Intrnational Journal of Mathmatical Analysis Vol. 8, 014, no. 49, HIKARI Ltd, An Approximation to th Boussinsq Equations Cristian-Fabian Baron-Prtuz Faculty of Exact and Natural Scincs, Univrsity of Cartagna Campus San Pablo, Avnu of Consulado Cartagna d Indias, Bolivar, Colombia Ana-Magnolia Marin-Ramirz Faculty of Exact and Natural Scincs, Univrsity of Cartagna Campus San Pablo, Avnu of Consulado Cartagna d Indias, Bolivar, Colombia Rubn-Dario Ortiz-Ortiz Faculty of Exact and Natural Scincs, Univrsity of Cartagna, Campus San Pablo, Avnu of Consulado Cartagna d Indias, Bolivar, Colombia Copyright c 014 Cristian-Fabian Baron-Prtuz, Ana-Magnolia Marin-Ramirz and Rubn-Dario Ortiz-Ortiz. This is an opn accss articl distributd undr th Crativ Commons Attribution Licns, which prmits unrstrictd us, distribution, and rproduction in any mdium, providd th original work is proprly citd. Abstract W study th systm of typ Boussinsq which modls th movmnt of a wav in th surfac of a fluid in a channl. W found solutions by using Fourir transform and Numann sris. Mathmatics Subjct Classification: 35Q35 Kywords: Boussinsq quations, Numann sris

2 434 C.F. Baron, A.M. Marin and R.D. Ortiz 1 Introduction In [], travling-wav solutions of Boussinsq systms was studid. It was constructd approximat and xact solutions for Boussinsq quations using homotopy prturbation Pad tchniqu in [5]. Mthods and applications to watr wavs wr studid in [6] and [3]. In [1] was tratd Boussinsq Equations. In papr [4] was obtaind th solitary wav solutions of two diffrnt forms of Boussinsq quations. In this papr w study nw approximatd solutions of th Boussinsq quations for a small paramtr α > 0 subjct to th condition η t + ((1 + αη)u) ξ = 0, (1) u t + η ξ + α (u ) ξ β 3 u ξξt = 0, () u(ξ, 0) = f(ξ), η(ξ, 0) = η 0 (ξ). (3) Exact solution in th nonlinar cas Dnoting th Fourir transform of a function g(ξ, t) with rspct to ξ by ĝ(k, t) = g(ξ, t) ikξ dξ Taking th Fourir transform in (1)-(3) w obtain ˆη t + ikû + αikˆη û = 0, ξ R, t 0, (4) (1 + β 3 k )û t + ikˆη + αû iqû = 0, (5) subjct to th condition û(k, 0) = f(k), ˆη(k, 0) = ˆη 0 (k). (6) Taking th drivativ with rspct to t in (5) w obtain (1 + β 3 k )û tt + ikˆη t + α(û iqû) t = 0, (7) Multiplying by ik to quation (4) and adding to quation (7) w obtain (1 + β 3 k )û tt + k û + αk ˆη û + α(û iqû) t = 0 (8)

3 An approximation to th Boussinsq quations 435 Dividing by k w hav whr th solution is (1 + β 3 k ) k û tt + û + αˆη û + α k (û iqû) t = 0 (9) ik t ik t û(k, t) = c 1 β 3 k + c β 3 k ikt + α t ikt t i + α β 3 k 0 Dividing by ik in (5) w obtain 0 i 1 + β 3 k f(τ) 1 + β 3 k f(τ) ikτ ikτ. (10) f(t) = ˆη û + 1 k (û iqû) t (11) Convoluting with û ˆη = 1 ik ((1 + β 3 k )û t + αû iqû) (1) ˆη û = 1 ik ((1 + β 3 k )û t + αû iqû) û Thn f(t) dpnds on û and from (10) whr ik t ik t û(k, t) = c 1 β 3 k + c β 3 k + αt û (13) ikt T û = t ikt t i + β 3 k 0 0 i 1 + β 3 k f(τ) 1 + β 3 k f(τ) ikτ ikτ

4 436 C.F. Baron, A.M. Marin and R.D. Ortiz For shallow watr wavs of small amplitud w hav that α and β ar vry small whr α is rlatd with nonlinar trm and β with rspct to th disprsiv trm and thy ar dimnsionlss paramtrs. Taking α 0 and using th fact that (1 αt ) 1 thr xists, thn applying Numann sris w hav (1 αt ) 1 = α n T n. Thn from (13) û(k, t) = n=0 n=0 Using th invrs Fourir thorm w obtain u(ξ, t) = 1 α n T n û(c 1 π n=0 α n T n ik t ik t û(c 1 β 3 k + c β 3 k ) (14) ik t + c ik t ) ikξ dk (15) This is an approximation of u as α 0. If w tak th drivativ with rspct to t in this quation w obtain ik ik û t (k, t) = c 1 t β 3 k + c ik 1 + β 3 k 1 + β 3 k + ( α n T n ik t ik t û(c 1 β 3 k + c β 3 k )) t n=1 ik t And thn rplacing this into (1) w gt ˆη = c β ik t 3 k β 3 k c 1 + β ik t 3 k β 3 k + O(α) (16) Now using th initial conditions (6) w obtain c 1 = f(k) c = f(k) W hav provd th following thorm ˆη 0 (k) + + O(α) 1 + β3 k ˆη 0 (k) + O(α) (17) 1 + β3 k Thorm.1. Th solution of th problm (1) - (3) is givn by (14), (16) and (17). Acknowldgmnts. Th authors xprss thir dp gratitud to Univrsidad d Cartagna for partial financial support.

5 An approximation to th Boussinsq quations 437 Rfrncs [1] J.L. Bona, M. Chn and J.C. Saut, Boussinsq quations and othr systms for small-amplitud long wavs in nonlinar disprsiv mdia. I: Drivation and Linar Thory, J. Nonlinar Sci, 1 (00), [] M. Chn, Solitary-wav and multipulsd travling-wav solutions of Boussinsq systms, Appl. Anal., 75, (000) [3] H. Hayashi, E. Kaikina, P. Naumkin and Shishmarv, Asymptotics for Dissipativ Nonlinar Equations, Lctur Nots in Mathmatics, 1884, Springr Vrlag, Brlin, 006. [4] A.J.M. Jawad, M.D. Ptkovic, P. Lakta and A. Biswas, Dynamics of shallow watr wavs with Boussinsq quation, Scintia Iranica., 0(1) (013), [5] M.M. Mousa, and A. Kaltayv, Constructing approximat and xact solutions for Boussinsq quations using homotopy prturbation Pad tchniqu, World Acadmy of Scinc, Enginring and Tchnology, 38 (009), [6] G.B. Whitham, Variational Mthods and Applications to Watr Wavs, Procdings of th Royal Socity of London. Sris A. Mathmatical and Physical Scincs, (1967), 6-5. Rcivd: Sptmbr 1, 014, Publishd: Octobr: 7, 014